{"paper":{"title":"Local compactness and nonvanishing for weakly singular nonlocal quadratic forms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Sven Jarohs, Tobias Weth","submitted_at":"2018-11-30T15:43:59Z","abstract_excerpt":"In this work we study a class of nonlocal quadratic forms given by \\[ \\mathcal{E}_j(u,v)=\\frac{1}{2}\\int_{\\mathbb{R}^N}\\int_{\\mathbb{R}^N}(u(x)-u(y))(v(x)-v(y))j(x-y)\\ dxdy, \\] where $j:\\mathbb{R}^N\\to[0,\\infty]$ is a measurable even function with $\\min\\{1,|\\cdot|^2\\}j\\in L^1(\\mathbb{R}^N)$. Assuming merely $j\\notin L^1(\\mathbb{R}^N)$, we show local compactness of the embedding $\\mathcal{D}^j(\\mathbb{R}^N)\\hookrightarrow L^2(\\mathbb{R}^N)$, where $\\mathcal{D}^j(\\mathbb{R}^N)$ denotes the space of functions $u\\in L^2(\\mathbb{R}^N)$ with $\\mathcal{E}_j(u,u)<\\infty$. Using this local compactness,"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.12850","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}